(w, a) ==/*«, V 154 XIII INTEGRATION each integer m ;> 1 we have (same method). (e) Put k(u) = sup h(u, oc), where the supremum is taken over all finite partitions of at X into u-integrable subsets. The number h(u) is called the entropy of u (relative to the measure /*). Show that, for each integer m ^ 1, we have h(um) = mh(u). (Remark that h(um, a) g h(um, V w~J(a) ) = w/z(w, a).) \ '-° / If moreover u is bijective and u"1 is ^-measurable, then hfa-1) = h(u). (f) Show that A(lx) = 0, where lx is the identity mapping of X. Deduce that if u* = ix for some integer p > 0, then h(u) = 0. 10. INTEGRALS OF VECTOR-VALUED FUNCTIONS Consider first a mapping f of X into a/w'/tf-dimensional real vector space m E. Let (ei)1^f^m be a basis of E, and put fto^^ffcfa for each xeX. Then f is said to be integrable (with respect to JJL) or ^integrabk if each of the real- valued functions ft is integrable, and we define (13.10.1) f J It is immediately checked that this definition is independent of the basis chosen. If ||z|| is a norm defining the topology of E, then we have the follow- ing criterion : (13.10.2) A mapping f : X ->E is integrable if and only iff is measurable and J *j|f(x)|| djjt(x) < + oo. The function x\-^> ||f(x)|| is then integrable. For f is measurable if and only if the ft(l ^i^ri) are measurable, by (13.9.6); also there exist two constants a, b such that by (5.9.1). The result now follows from (13.9.13) and (13.9.6). In particular, a complex function/ on X is integrable if and only if+n-l \